See [1]}, [2]} for details about \(S(z)\) . We remark that
\(z\mapsto \overline{S(z)}\)  is the (local) anti-conformal reflection map in \(\partial \Omega \) 
and that \(S^{\prime }(z)=T(z)^{-2}\) , where
\(T(z)\)  is the positively oriented and holomorphically extended unit tangent vector on \(\partial \Omega \) .
